Monday, August 14, 2017

The Mayan Number Activity

While I was living in Guatemala this summer, I was introduced to Mayan numbers. Mayan numbers are interesting because they are base 20 and they are written vertically.

I knew I wanted to use Mayan numbers for an activity in my algebra classes, but wasn't sure where to go with it. A code-breaking activity seemed like a natural fit, but what would the motivation be for breaking the code? And how could this activity reinforce mathematical thinking?

As I started thinking about the first day of class, I realized it could be cool to "introduce myself" to the students using Mayan numbers. There are always a few things students REALLY want to know about me. What if I gave them the most common questions about myself AND gave them the answer in Mayan code? I thought this would be a great way to reinforce my belief that the answer is the least important part of the problem, while also giving students a chance to get to know me.

I put the activity together, and was pretty happy with the results. I wrote questions that were interesting (my age, the number of tattoos I have). I also used Mayan numbers that were 'easy' to figure out like 6 and 7 while also giving Mayan numbers that were challenging like 25 and 38. Basically creating a low floor, high ceiling task. In Mayan numbers, 25 looks just like the number 6, except the 'dot' in 25 is much higher--in Mayan numbers the height represents place-value. I liked what I had, but something was missing.

So, like I also do when I get stuck, I reached out to two colleagues for their advice. One colleague had recently gone to a workshop with Dan Meyer where he talked about a Goldilocks Guess (too high, too low, and a range of numbers that might make sense). I realized this was the missing piece for this activity!

Welcome to My Class

I've never been the type of teacher to read the syllabus on the first day. But I usually spend about 20 minutes on classroom routines and norms and then 20 minutes on a get-to-know-you, non-math activity like "stand up if...". [At my school the first day of class has 49 minute periods]

This year, I skipped all that! As students walked in, I let them sit where they wanted. (This was NOT a problem at all, but I will admit, I was scared about self-seating).

There were about 10 minutes of procedural stuff I wanted to go over. I did that. Then I showed the students this slide.

Then I asked students to take the following cards out of the group folder and to give one card to each group member. (students sit in groups of 4).

First, I asked students to think to themselves about their card for one minute. I showed students the following slide.

Then, I asked students to talk to their partner. The purpose was twofold: introduce themselves to their partner and also start to compare the different cards. I showed them this slide and let them talk for about two minutes.

Finally, I had the students stand up and huddle as a team of 4. I asked students to share their card with their group members and then try to order the cards from least to greatest. I also told groups they could ask me one question to help break the code. I would NOT answer a question about the code, but I would answer a question they had in code. For example, if they wanted me to write their age in code, I would do that. Groups worked for about 10 minutes. And they had AMAZING conversations.

As students worked, my student teacher and I circulated to look for interesting student work and interesting (but not necessarily correct) student thinking. I want to shout out Ms. D, my student teacher, for finding AMAZING examples of student thinking and for warmly inviting students to come up to the front of the class to share their thinking on the first day of school!

Ms. D did the following:

  1. She first called up a female student that figured out that the bar represented five and the dot represented one. The student talked about the question her group asked Ms. D (how to write the number 13 in code) and how she thought backwards to figure out what the bar and dot represented.
  2. Ms. D then called up a male student that built on the previous work. He knew that a bar was five and a dot was one, but couldn't figure out what that "floating dot" represented, but he knew that the floating dot was significant. I gave him a shout out for sharing a point of confusion. Something students are scared to do on the first day of class! For my age, he knew that the lower part of the number was 18, but he also recognized that I'm not 18. He just couldn't quite get the floating dot. 
We paused as a class to talk about the floating dot. I gave students time to talk to their partner. I asked them: What could the floating dot represent? Is it important? As I circulated, I heard one student share that he thought the floating dot was a 20, but couldn't quite explain why besides given the context of the questions, 20 made sense. My other class thought the floating dot was 15 for the same reason. Another student talked to his partner about an "invisible line". Basically the vertical division that gives the number a place value.  As I called on students to share their thinking with the class, I could see the light bulbs going off around the room. Wow!! I couldn't have been happier with this rich class discussion ON THE FIRST DAY OF SCHOOL!! 

Once they put it all together, I showed them the answers. The bell rang. Students left. And Ms. D and I were very, very happy about our first day of class! :) 

Monday, August 7, 2017

Are You There, Rick? It's Me, Your Teacher Voice

This post is about changing the culture around homework in my classroom from procedural practice to growth mindset. But this post is also about having the courage to listen to, and act on, my teacher voice. 

Background (the why)

If you're not interested in the why...skip down to 'the how'. 

In my first year of teaching, homework took up an obscene amount of my time. I was teaching algebra 2 and each of my classes had between 35 and 40 students. I gave daily homework for each class. Usually a worksheet (front and back) created by me or one of the other alg2 teachers. The homeworks we created weren't bad, but they were procedural--aka drill and kill.

Not much changed in my second year of teaching with regards to homework. I still taught alg2. I still had huge classes. And grading homework still took up an inordinate amount of my time as well as the students' time.

Even in my first year, I could hear my teacher voice whispering that time spent on homework wasn't an effective use of time. My time was wasted because I wasn't using the homework in a meaningful way: it wasn't a formative assessment and the time I spent grading and/or making homework was time I wasn't using to reflect on and refine my teaching practice.

The students time was wasted because homework did not create opportunities for students to understand mathematics. Moreover, the students that knew how to 'do school' did the homework. And the students that struggled with 'doing school' didn't do the homework; for these students, not doing homework hurt their grade--and often that low grade wasn't a reflection of their understanding of mathematics. Not only was homework a waste of time, it was an equity issue!

In my second year, I kept hearing my teacher voice say that homework wasn't an effective use of our classroom time. But I didn't have the courage to act on my voice. All the other alg2 teachers gave homework; I was new-ish (in my 2nd year) and didn't feel comfortable going rogue. In hopes of finding a solution, I talked with the other alg2 teachers about homework. Most of them graded homework by using class time to go around the class, look at student's homework, and stamp it (or not) for completion. While the teacher stamped, the students did a procedural warm-up activity. At end of the week or unit, the teacher would collect the stamp sheet and use that to give the students a homework grade. I tried stamping and failed miserably at it. The students didn't do so well either! And, again, my teacher voice told me this wasn't an effective use of our time.

Over winter break that second year I thought a lot about homework. I realized I needed to blow up my current system and start from scratch. And to do that, I had to find the courage to listen to my teacher voice. Here is what I did.

Growth Mindset Makeover (the how)

I decided to change the culture around homework in my classroom. Rather than homework being procedural practice, I decided to make homework about growth mindset. To do this I decided I would no longer correct the math content of their homework. Rather, I would have the students grade their own math work and I would grade the quality of the corrections to their own work. I created this rubric.
For my mainstream class

For my algebra class for English Language Learners (don't judge my awful Spanish!!)

What Does the Homework Routine Look Like in My Classroom

  1. As we begin class, students take out their homework. I project a homework key and students grade their homework using a different color pen (very important). 
  2. As students correct their homework, I circulate to look for a few different things: I formatively assess their work. And I also give students feedback on the quality of their corrections--in other words, I guide the development of the homework culture.  I explain this in more detail below. 
  3. This entire homework correction process takes less than 10 minutes of class time. However, my school is on a block schedule so a class 'period' is around 85 minutes in my school (I see students every other day). 
  4. Students turn in their work. I don't grade the content, I grade the quality of the corrections to their homework. 

Using Homework as a Formative assessment

Circulating while students correct their own homework allows me to see if there is a common error. Often, students are shy to ask questions about errors, especially in the beginning of the year. Thus, I can formatively assess their work via the comments. If there is a common error that I feel needs to be addressed, I will stop the class and take time for a whole class discussion about that particular error. I typically say something like: “I LOVE this error, I’ve seen several of you make this error on this homework. Let’s pause to discuss the error. Talk to your partner. What was this student thinking when they made this error?”

This is informative for my practice. If everyone is making the same error, it is information I can use to inform my next steps. Do I need to reteach? Assign certain students to the intervention period to work with me in a smaller group? Looking at their comments allows me to assess them real-time and adjust my lesson as needed.

Guiding the Homework Culture

My homework system is often a shift in culture for my students. In the past, homework was either stamped for completion or graded by the teacher as correct/incorrect. Thus, while I circulate, I can give feedback on the quality of their corrections. And my feedback guides students towards my vision for homework culture. “Wow! What a specific and reflective comment!” or “What was the error in your thinking? Please write a more specific and reflective comment on your work.” Especially in the beginning of the year, students tend to write superficial comments like “I got this wrong” or “try harder”. I gently let them know these are not the type of comments I expect. I expect comments that are reflective and specific.

I will also use this time as an opportunity to raise student status. If there is a student I know struggles with math or has a mathphobia, and I see that student write a particularly reflective, specific comment, I will tell the student that I LOVE their comment and I will ask them if I can take a picture of it. If the student says yes, and they always do, I will take out my phone and take a picture of the work and the comment. This is powerful. I’m not acknowledging the student with the least errors; I’m not acknowledging the student that did the ‘best math’; I’m acknowledging the student that was the most thoughtful, specific and reflective about the errors in their work. This is a great way to build a growth mindset and a great way to address the ever present mathphobia that students have when it comes to making mistakes in math.

Circulating also allows me to catch students that are trying to ‘cheat the system’. Yes, unfortunately, this happens—especially in the beginning of the year. Students will bring in a blank homework and ‘correct it' by filling in the solutions (still using a different color pen!). As I circulate, I can easily spot this as it happens and have a quick, one-on-one performance conversation with the student. “This homework is blank, so there is nothing for you to correct. Put this away, finish the homework outside of class, and you can come correct it outside of class time and turn it in late for partial credit.”

My Mistakes In Introducing This Process

My biggest mistake in implementing a change in homework policy and attempting to shift the culture around homework was not honoring the time investment and communication needed to shift classroom culture. 

At my first attempt in rolling this out to the students, I introduced the rubric, explained what I wanted and expected the students to immediately understand and connect to my vision.  That didn't happen! 

Now I realize changing culture, especially around something with a lot of baggage like homework, takes weeks (or months) and requires a lot of feedback and guidance from me. Besides the feedback I give while circulating, I also show students examples of ideal and not ideal corrections to their work. I will scan their corrected work and anonymously project corrections that meet my expectation as well as corrections that do not meet me expectation. In the beginning of the year, I take a minute or two to do this daily. Later, I usually show them ideal/not ideal examples once or twice a month. 

Here are examples of not ideal corrections. I show these to students and ask them to discuss the quality of the corrections. Are the reflections reflective and specific? If not, what could be improved in terms of being reflective and specific?

Ideal Corrections (specific, reflective)


Not Ideal Corrections (vague, superficial)


Listening to my Teacher Voice

It took a year and a half, but when I finally listened to my teacher voice, it was transformative for my teaching practice. As I reflect on the process used to come to my new homework policy, I realize that one of the things that makes education challenging is that often there is no one 'right way'. As an educator, I can talk to and listen to other teachers, I can also stay up-to-date on research and professional articles; however, those sources of data can only serve to inform my practice--ultimately, the day-to-day decisions that happen in my classroom need to resonate with me, my students, my context, and my vision for my classroom culture. And that means I need to have the courage to listen to my teacher voice. Even when it seems like my voice is in the minority, I need to understand that my opinion and my voice matter. I just need to make sure I'm listening!

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